Determinantal Martingales and Interacting Particle Systems

TL;DR

This paper introduces determinantal martingales and demonstrates how they characterize determinantal processes, especially in noncolliding diffusion processes, providing a unified framework for understanding their correlation structures.

Contribution

It establishes that if an interacting particle system has a determinantal-martingale representation, it is a determinantal process, and explains the determinantal nature of noncolliding processes via harmonic transforms.

Findings

Determinantal martingales characterize determinantal processes.

Noncolliding diffusion processes are determinantal due to harmonic transforms.

A new perspective on quantum Toda lattice related processes is provided.

Abstract

Determinantal process is a dynamical extension of a determinantal point process such that any spatio-temporal correlation function is given by a determinant specified by a single continuous function called the correlation kernel. Noncolliding diffusion processes are important examples of determinantal processes. In the present lecture, we introduce determinantal martingales and show that if the interacting particle system (IPS) has determinantal-martingale representation, then it becomes a determinantal process. From this point of view, the reason why noncolliding diffusion processes and noncolliding random walk are determinantal is simply explained by the fact that the harmonic transform with the Vandermonde determinant provides a proper determinantal martingale. Recently O'Connell introduced an interesting IPS, which can be regarded as a stochastic version of a quantum Toda lattice. It is a geometric lifting of the noncolliding Brownian motion and is not determinantal, but Borodin and Corwin discovered a determinant formula for a special observable for it. We also discuss this new topic from the present view-point of determinantal martingale.